Convex optimization of coincidence time resolution for a high-resolution PET system

Abstract

We are developing a dual panel breast-dedicated positron emission tomography (PET) system using LSO scintillators coupled to position sensitive avalanche photodiodes (PSAPD). The charge output is amplified and read using NOVA RENA-3 ASICs. This paper shows that the coincidence timing resolution of the RENA-3 ASIC can be improved using certain list-mode calibrations. We treat the calibration problem as a convex optimization problem and use the RENA-3's analog-based timing system to correct the measured data for time dispersion effects from correlated noise, PSAPD signal delays and varying signal amplitudes. The direct solution to the optimization problem involves a matrix inversion that grows order (n(3)) with the number of parameters. An iterative method using single-coordinate descent to approximate the inversion grows order (n). The inversion does not need to run to convergence, since any gains at high iteration number will be low compared to noise amplification. The system calibration method is demonstrated with measured pulser data as well as with two LSO-PSAPD detectors in electronic coincidence. After applying the algorithm, the 511 keV photopeak paired coincidence time resolution from the LSO-PSAPD detectors under study improved by 57%, from the raw value of 16.3 ±0.07 ns full-width at half-maximum (FWHM) to 6.92 ±0.02 ns FWHM ( 11.52 ±0.05 ns to 4.89 ±0.02 ns for unpaired photons).

Fig. 1

Stepping through threshold values shows the shape of the input signal to the leading edge discriminator. Lowering the threshold causes the negative pulse to trigger the comparator at a later time.

Fig. 2

The quadrature timing signals, U and V, when plotted against each other for 54098 pulses, form a circle. The circle has a radius of and average of 2438 counts and a standard deviation of 9 counts.

Fig. 3

Pulser timing histogram from channel 16 to the other 35 channels. Each line shows a different channel. A range of greater than 25 ns exists across the channels.

Fig. 4

Plots of the quadrature timing signals against the calculated time difference show a strong correlation which can be corrected to improve the pair coincidence timing resolution.

Fig. 5

Digitized PSAPD pulse shapes. The PSAPD signals used for timing were measured using free-running ADCs [24]. The delay in arrival time between events in crystals near the center of the PSAPD compared to the corner of the PSAPD is visible.

Fig. 6

Per crystal timing delay caused by the PSAPD resistive sheet. The gray scale units are nanoseconds. The bottom right corner is the crystal used as the timing reference. Timing for crystals on a single PSAPD is achieved using timing measurements from only one RENA-3 channel.

Fig. 7

Two 8×8 arrays of 1 × 1 × 1 mm³ crystal elements coupled to PSAPDs are placed edge-on, 6 cm apart. A ²²Na point source is placed in between the two crystal arrays. The crystal flood histogram for each array used for crystal position gating is shown below the crystal.

Fig. 8

Timing histograms from ASIC channel 16 to the other 35 channels using pulser data. Each line shows a different channel. Removal of channel skew through delay correction improved the coincidence time resolution from an uncorrected 25 ns range to 4.39 ± 0.04 ns FWHM.

Fig. 9

Timing histograms from ASIC channel 16 in reference to the other 35 channels using pulser data. Each line shows a different channel. Correction for quadrature timing correlated noise has a 22% improvement over just delay correction to achieve 3.37±0.02ns.

Fig. 10

Measured pulser coincidence time resolution for all pairings of channels in the ASIC. The whiter shade signifies a greater FWHM. Units are FWHM in nanoseconds. The diagonal of zeros down the center is due to no data for a channel being paired with its self.

Fig. 11

The individual channel (unpaired) timing resolutions for pulser data. The unpaired channel time resolution varies from 1.38 ± 0.02 ns FWHM to 4.48 ± 0.02 ns FWHM.

Fig. 12

Predicted coincidence time resolution between all pairings of channels for pulser data. The whiter shade signifies a greater FWHM. Units are FWHM in nanoseconds. The diagonal, where channel numbers of each axis are the same, represents the time resolution if it was possible to measure two coincident events on the same channel.

Fig. 13

An energy spectrum from an 8 × 8 array of l×l×l mm LSO-PSAPD detector. The gain is calibrated per crystal and the resulting energies combined to form the spectrum of the whole detector. The energy resolution is l0.9±0.1% FWHM and the lutetium X-ray escape energies are visible in the lower energy side of the photopeak.

Fig. 14

Timing histograms of coincidence timing between LSO-PSAPD detectors of Fig. 7. From top to bottom: No correction, FWHM 16.38 ± 0.05 ns; ASIC inter-channel delay correction, FWHM 8.42 ± 0.03 ns; inter-channel delay, signal amplitude and quadrature timing correction, FWHM 6.92 ± 0.02 ns; and the improvement of timing resolution with each calibration.

Fig. 15

The optimum delay adjustment for the two 8×8 crystal arrays coupled to PSAPDs used in the coincident setup of Fig. 7. The gray scale is in nanoseconds. There is up to 20 ns delay across the PSAPD sensitive surface, and a 10 ns delay between the two arrays.

Fig. 16

A closer look at the coincidence time resolution over all 128 1×1×1 mm³ crystal elements in the two LSO-PSAPD detectors shows two side lobes. These lobes are the result of multiple interactions in a single array.

Fig. 17

The top two images show the resulting crystal flood histograms in each of the two opposing detectors (Fig. 7) when events are gated around the negative side lobe from the coincidence time histogram of Fig. 16. Individual crystals are visible in the left crystal array, but blurred in the right crystal array. The bottom two images show that for gating around the positive side lobe, where the reverse is true. These data indicate that the side lobes are the result of multiple interactions in a single array.

Fig. 18

Histograms of the backprojected lines of response from the coincidence data with a 0.250-mm-diameter point source with a 0.33 mm pixel/bin size. The bottom plot is a histogram of the lines of response taken from a vertical profile through the center of the image. The raw FWHM is 0.90±0.05 mm (before beam size deconvolution).